Learning a robust shape parameter for RBF approximation

TL;DR

This paper introduces a novel, data-driven approach to select the shape parameter in RBF interpolation, optimizing stability and accuracy by controlling the condition number of the interpolation matrix, with adaptive learning strategies.

Contribution

It proposes a new optimization and learning-based method for choosing the shape parameter in RBFs, ensuring stable and accurate interpolation.

Findings

The method effectively controls the condition number of the interpolation matrix.

Numerical tests show improved stability and accuracy in interpolation tasks.

The approach enhances RBF-FD methods in one and two dimensions.

Abstract

Radial basis functions (RBFs) play an important role in function interpolation, in particular in an arbitrary set of interpolation nodes. The accuracy of the interpolation depends on a parameter called the shape parameter. There are many approaches in literature on how to appropriately choose it as to increase the accuracy of interpolation while avoiding instability issues. However, finding the optimal shape parameter value in general remains a challenge. In this work, we present a novel approach to determine the shape parameter in RBFs. First, we construct an optimisation problem to obtain a shape parameter that leads to an interpolation matrix with bounded condition number, then, we introduce a data-driven method that controls the condition of the interpolation matrix to avoid numerically unstable interpolations, while keeping a very good accuracy. In addition, a fall-back procedure is proposed to enforce a strict upper bound on the condition number, as well as a learning strategy to improve the performance of the data-driven method by learning from previously run simulations. We present numerical test cases to assess the performance of the proposed methods in interpolation tasks and in a RBF based finite difference (RBF-FD) method, in one and two-space dimensions.